Exploring the Bose Hubbard Model
Akshay Shankar
2022-09-24
Interacting bosons in a lattice
\[H = -t\sum_{\langle i, j \rangle}
a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)\]
Phases of the BHM
Mott Insulator
\[\small H = \frac{U}{2}\sum_i n_i(n_i - 1)
\hspace{0.5cm}\longrightarrow\hspace{0.5cm} |\Psi_{MI}\rangle =
\prod_{i=1}^M |n\rangle\]
Superfluid
\[\small H = -t\sum_{\langle i, j \rangle}
a_i^{\dagger}a_j\hspace{0.5cm}\longrightarrow\hspace{0.5cm}|\Psi_{SF}\rangle=
\frac{1}{N!} (\sum_{i=1}^M a_i^{\dagger})^N |{0}\rangle
\hspace{0.5cm}\]
Exact Diagonalization
Consider N bosons in M lattice sites.
- \[\text{Enumerate basis set:
}\hspace{0.5cm} |n_1, n_2, ..., n_M\rangle \hspace{0.5cm} s.t.
\hspace{0.5cm} \sum_{i=1}^M n_i = N\]
- \[\text{Construct hamiltonian matrix:
}\hspace{0.5cm} \langle n_1, n_2, ..., n_M| \hspace{0.1cm}\hat{H}
\hspace{0.1cm}| n_1', n_2', ..., n_M'\rangle\]
- \[\text{Diagonalize the hamiltonian: }
\hspace{0.5cm} H|\psi\rangle = E|\psi\rangle\]
Results (ED)
Sparsity of the Hamiltonian
10 bosons; 10 lattice sites.
Dimensionality scaling
\[\small\text{Dim }(\mathcal{H}) =
\frac{(N+M-1)!}{N!(M-1)!}\]
Order of Magnitude
Mean Field Theory
\[\small \hat{a}_i = \Psi +
\delta\hat{a}_i \hspace{1cm} \Bigg |\hspace{1cm}\mathcal{O}(\delta
\hat{a}_i ^2) \approx 0\]
\[\small \underbrace{H =
-t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i
n_i(n_i - 1)}_{\text{coupled lattice sites}}
\hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi \} =
\sum_i-zt \cdot (\Psi^*a_i + \Psi a_i^{\dagger} - |\Psi|^2) +
\frac{U}{2}n_i(n_i -1)}_{\text{de-coupled lattice sites}}\]
Solution (MFT)
To find the equilibrium state of the system, we must minimize
\(G = H - \mu N + TS\).
\[\frac{\partial \langle G\rangle}{\partial \Psi} =
\frac{\partial \langle H - \mu N\rangle}{\partial \Psi}\Bigg|_{T=0K} =
0\]
\[ \Psi = \langle \psi_{gs} |\hat{a} |
\psi_{gs} \rangle\]
\[\text{Minimize G} \equiv \text{Self-consistently
diagonalize } H\{\Psi\}\]
Algorithm (MFT)
Results (MFT)
Results (MFT, contd.)
Cluster Mean Field Theory
\[\small \underbrace{H = -t\sum_{\langle
i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i -
1)}_{\text{coupled lattice sites}}
\hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi_i \} =
\sum_C H_{exact} + \sum_{C, C'}H_{MFT}\{ \Psi_i
\}}_{\text{de-coupled clusters of sites}}\]
Results (CMFT)
Dipolar bosons in a lattice
\[H = -t\sum_{\langle i, j \rangle}
a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1) + V \sum_{\langle i, j
\rangle} n_i n_j\]
Solution (MFT)
\[\small H = -t\sum_{\langle i, j \rangle}
a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1) + V\sum_{\langle i, j
\rangle} n_i n_j\] \[\hat{n}_i =
\rho_i + \delta \hat{n}_i\hspace{0.5cm}\Bigg\downarrow
\hspace{0.5cm}\hat{a}_i = \Psi_i + \delta\hat{a}_i\] \[\small H_A \{\Psi_A, \Psi_B,\rho_A, \rho_B \} =
-zt \cdot (\Psi_B^*a_A + \Psi_B a_A^{\dagger} - \Psi_A^*\Psi_B) +
zV\cdot(\rho_Bn_A - \rho_A\rho_B) + \frac{U}{2}n_A(n_A -1) \\
\small H_B \{\Psi_A, \Psi_B,\rho_A, \rho_B \} = -zt \cdot (\Psi_A^*a_B +
\Psi_A a_B^{\dagger} - \Psi_B^*\Psi_A) + zV\cdot(\rho_An_B -
\rho_B\rho_A) + \frac{U}{2}n_B(n_B -1)\]
\[\small H \{\Psi_A, \Psi_B,\rho_A, \rho_B\}=
\sum_{i \in A} H_i + \sum_{j \in B} H_j\]
Solve self-consistently: \(\hspace{0.5cm}\Psi_i = \langle \psi_{gs,
i}|\hat{a}_i |\psi_{gs, i}\rangle \hspace{0.5cm};\hspace{0.5cm} \rho_i =
\langle \psi_{gs, i}|\hat{n}_i |\psi_{gs, i}\rangle \hspace{0.5cm};
\hspace{0.5cm} i \in \{A, B\}\)
Results (MFT)
Results (MFT, contd.)
